Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials

Abstract

Abstract. We study the specialization of the type A nonsymmetric Macdonald polynomials at t " 0 based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of the Kostka-Foulkes polynomials.

Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ and $$q=0$$

“…, x n ] and is a natural generalization of the elementary symmetric functions in the same manner the key polynomials extend the family of Schur polynomials. For example, in a recent paper [3], it is proved that E λ (x; 1, 0) expands positively into key polynomials, where the coefficients are given by the classical Kostka coefficients. This generalizes the classical result that elementary symmetric functions expand positively into Schur polynomials.…”

“…In particular, these expand positively into key polynomials with Kostka-Foulkes polynomials (in q) as coefficients. There are representation-theoretical explanations for these expansions, as well, see [2,3] and references therein for details.…”

“…Our results are in some sense optimal: for general compositions λ, it happens that E σ λ (x; 1, t) cannot be factorized further. For example, Mathematica computations suggest that E (3,1,5,2,4) (0,2,3,1,0) (x; 1, t) and E (3,1,5,2,4) (0,1,1,1,0) (x; 1, 0) are irreducible.…”

“…The rows with labels 2 and 3 constitute a monotone pair and can be obtained using (5.2), which explains theπ 2 -arrow. Continuing on withπ 1 followed byθ 2 leads to an augmented diagram without any monotone pairs, so A 1342 3102 (x; t) = x (3,2,1,0) . Finally, following the arrows yields the operator expression:…”

A Major-Index Preserving Map on Fillings

“…Note that a recent combinatorial proof of the key expansion of Eα(x; q, 0) using weak dual equivalence was given in [Ass17].…”